Mixed convection flow of an electrically conducting viscoelastic fluid past a vertical nonlinearly stretching sheet

The study of hydromagnetic mixed convection flow of viscoelastic fluid caused by a vertical stretched surface is presented in this paper. According to this theory, the stretching velocity varies as a power function of the displacement from the slot. The conservation of energy equation includes thermal radiation and viscous dissipation to support the mechanical operations of the heat transfer mechanism. Through the use of an adequate and sufficient similarity transformation for a nonlinearly stretching sheet, the boundary layer equations governing the flow issue are converted into a set of ordinary differential equations. The Keller box technique is then used to numerically solve the altered equations. To comprehend the physical circumstances of stretching sheets for variations of the governing parameters, numerical simulations are made. The influence and characteristic behaviours of physical parameters were portrayed graphically for the velocity field and temperature distributions. The research shows that the impact of the applied magnetic parameter is to improve the distribution of the viscoelastic fluid temperature and reduce the temperature gradient at the border. Temperature distribution and the associated thermal layer are shown to have improved because of radiative and viscous dissipation characteristics. Radiation causes additional heat to be produced in liquid, raising the fluid's temperature. It was also found that higher velocities are noticed in viscoelastic fluid as compared with Newtonian fluid (i.e., when K = 0).


Scientific Reports
| (2022) 12:14679 | https://doi.org/10.1038/s41598-022-18761-0 www.nature.com/scientificreports/ We concentrate on nonlinear stretching sheets in this study. As a result, external flow control mechanisms, such as a magnetic field, are required to assure the correct feature. One field of research that combines the mechanics of fluids with some electromagnetic aspects to describe the transport phenomena in an electrically conducting fluid is magnetohydrodynamic (MHD) fluids, a branch of fluid mechanics that involves the analysis of electrically conducting fluids. The performance of fluid flow and heat transfer phenomena is influenced, controlled, and regulated by the existence of a magnetic field inside the fluid. It also has a substantial impact on solar physics, bioengineering, heat and mass transfer, metal, high thermal plasmas, MHD power generators, MHD reactors, thermal insulation, turbines, nuclear reactors coolant, and electronic packages, among other fields 9 . The variation of radiation and heat generation/absorption on MHD viscoelastic fluid flow via a stretched surface were examined by Datti et al. 10 . According to their findings, the existence of a magnetic field in the fluid causes a high rate of heat transmission, which improves heat transfer at the surface. Similar research is conducted by Abel et al. 11 on the magnetohydrodynamic flow of viscoelastic fluid caused by a stretched surface. They noticed that as the magnetic parameter improves, the speed of the flow field reduces. This is brought on by a drag force that seeks to impede fluid flow and lower its velocity, known as the Lorentz force. Sharada 12 examined the effect of a magnetic field on a steady flow of viscoelastic fluid. The author reveals that the Lorentz force gains more strength with the increased values of a magnetic parameter. Kumar and Sivaraj 13 also examined the magnetohydrodynamic natural convective flow of Walters' B viscoelastic fluid over a flat and vertical cone contained in a porous media. In addition to exhibiting both viscosity and elasticity, one category of non-Newtonian fluids with dual effect features is viscoelastic fluid (i.e., heat transfer reduction and drag reduction properties) 14 , which contributes to a wide range of applications in the polymer industry. Manufacture of paper, fibre glass, dyes, films, extraction processes, copper wire thinning and hardening, synthetic fibre and plastic film processing, drawing of plastic sheet, cooling of metallic chips, and food processing are all included in this 15 . In the process engineering of oil reservoirs, bio-engineering, the chemical industry, and nuclear technology, viscoelastic fluid is also crucial. Physically, the behaviour of viscoelasticity helps in depressing or enhancing the rate of the heat transfer process. However, it relies on the kinematic properties of the velocity field being studied as well as the heat exchange direction. Studying viscoelastic fluid and transport of heat adjacent to the continuous moving surface is an essential area of research as it helps in determining the desired and favourable output in most industrial and engineering processes. For this purpose, Abel 16 carried out heat transfer analysis in electrically conducting flow of viscoelastic fluid induced via non-isothermal linear stretching surface in the presence of heat generation and found that the temperature distribution within the boundary region reduces with an upsurge in viscoelasticity. The free conductive flow of Walters' B viscoelastic fluid via movable flat plates and vertical cones in the presence of varying electric conductivity was also examined by Sivaraj and Kumar 17 . Their research revealed that the magnetic field and varied electric conductivity had a substantial impact on the viscoelastic flow.
Furthermore, the numerical solution of a viscoelastic fluid passing a flat plate was achieved by Maryam et al. 18 . They used a second-order parallelized finite volume method and noticed that the drag coefficient at the boundary of viscoelastic fluids is lesser as compared to that of Newtonian fluids. Similarly, the impact of a magnetic field on two kinds of viscoelastic fluids produced by a stretched sheet was theoretically examined by Cortell 19 . According to his observations, the viscosity is greater in the second-grade fluid when compared to Walter's B liquid. Mustapha 20 examined the analytical solution of viscoelastic fluid generated by a nonlinearly stretching sheet by employing the method of homotopy analysis. The impact of viscous dissipation on the viscoelastic fluid flow caused by a vertical nonlinear stretching sheet was quantitatively examined by Jafar et al. 21 while Seth et al. 22 investigated the influence of velocity slip on the Dufour and Soret phenomena of MHD viscoelastic fluid created by a nonlinearly stretched surface. It was found that the viscoelasticity parameter reduces the fluid's temperature. Recently, Megahed et al. 23 examined the variations of viscous dissipation and variable fluid properties on the transport phenomenon of magnetohydrodynamic viscoelastic fluid. They discovered that as the viscosity and viscoelasticity parameters increase, the sheet velocity also increases. The influence of chemical reaction on Buongiorno's nanofluid model of viscoelastic fluid via a nonlinear stretched surface has been quantitatively examined by Nadeem et al. 24 .
The thermophysical characteristics of fluid are subtly connected to the phenomena of heat transport systems. Indeed, the increasing demands to enhance heat dissipation, and cooling/heating processes as well as saving energy, time, and cost of production in the industrial system has been a challenge to many engineers and scientists. However, the analysis of boundary layer flow and heat transport, which happens when a heated surface moves continuously over a static fluid, has several technical uses in engineering and industrial production because it is crucial in determining the properties of the process's end product. This involves the process of extrusion of hot plastic and metals, continuous casting, hot rolling, and production of glass fiber and paper 25 . Convective heat transfer is usually the heat transfer process that exists in the presence of a fluid motion on a solid surface. Convective heat transfer is divided into two or three categories, which in turn aid the transfer of heat. The first category, which is usually called natural or free convection, arises due to the natural buoyancy forces (i.e. density gradient). The second category, which is usually called forced convection, arises due to the pressure differences and the last category occurs in a situation where the free convection is accompanied by the external force. In this case, mixed convection flow is the term for the type of heat transfer that results from the combined action of free and forced convection 26 .
Much emphasis has been paid to the investigation of mixed convective flow along a rigid plate lately because it is essential for a variety of applications, including the electronic cooling devices by fans, the cooling of nuclear reactors during an unexpected failure, the placement of heating systems in minimal-velocity environments, solar cells, and others. When the flow is horizontal, the buoyancy force is disregarded in the analysis of flow across hot surfaces. However, the buoyant force has a substantial impact on the flow field for vertical or inclined surfaces. To achieve this, Lloyd 28 probed the impact of varying fluid characteristics on the mixed convection heat transport via a nonlinearly stretching sheet. Soret and Dufour effects on the convective flow of viscoelastic fluid over a stretched sheet encased in a porous medium were also studied by Jena et al. 29 . Furthermore, Hayat et al. 30 also examined a viscoelastic Walters-B nanofluid mixed convection flow via a nonlinear vertical stretched surface with varying thickness.
The emission of energy in the form of electromagnetic waves is referred to as radiation. Through this method, energy can be conveyed as heat, energy, or light. Numerous fluid parameters are altered when such emission results from fluid flow 24 . The thermal radiation effect is important for adjusting a system's temperature, modifying the pace of heat transport, and managing the thermal boundary layer. To explore the influence of thermal radiation on the natural convective flow Powell-Eyring nanofluid via a cylinder, Kumaran et al. 31 conducted an investigation. They observed that a reduction in heat transfer processes results from an increase in the radiation parameter. Many studies, like Raja et al. 7 , Datti et al. 11 , and Ahmad et al. 15 , have considered the impact of radiant heat on boundary layer flow induced by a nonlinearly stretching sheet because of its significance in controlling and maintaining the system's temperature.
One must also consider the viscous dissipation effect in addition to the impacts of magnetic and thermal radiation on the flow field. The energy created by the work done as a result of frictional heating between fluid layers is known as viscous dissipation. By acting as an energy source, the heat energy generated modifies temperature distributions, which in turn affects fluid temperature and heat transfer rates. Greater gravitational fields, massive planets, denser space gases, and geological phenomena all result in viscous dissipation 32 . Several authors have considered viscous heating effects on boundary layer flows, such as Hsiao 33 , Reddy et al. 34 , Yaseen et al. 35 , and Jhahan et al. 36 .
The primary objective of the current study is to find a numerical solution for the flow of a viscoelastic fluid through a mixed convection boundary layer as a consequence of the cumulative impact of a uniform magnetic field, viscous dissipation, and thermal radiation via a vertical sheet that is not linearly stretching. By employing an unconditionally stable Keller box strategy 37 to analyse a group of linked nonlinear ordinary differential equations, the non-similar solutions of the modified equations are achieved numerically. This method has been widely used by many researchers like Kumaran et al. 38 and Hayath et al. 39 in dealing with parabolic differential equations. For more details see 40,41 . Governing equations. We take into account a viscoelastic fluid flowing in mixed convective two-dimensional magnetohydrodynamic flow across a stretched sheet, as seen in Fig. 1. The extrusion slit, where the sheet is drawn, serves as the system's point of origin. The x-axis is drawn along the continuously extending surface and faces forward. The sheet is parallel to the y-axis. The sheet is subjected to a perpendicular B(x) magnetic field. The sheet velocity, U w (x) = ax n is assumed to vary as a nonlinear function of the distance from the slit, where a > 0 is the stretching rate and n is a nonlinear stretching parameter. The temperature of the fluid, T w (x) = T ∞ + T 0 x 2n−1 is also considered as a nonlinear function of the distance from the slit, where T 0 is a positive constant and T ∞ is the ambient temperature of the fluid. Thus, using the conventional boundary layer approximation, the viscoelastic fluid equations governing the momentum and energy equations may be expressed as follows 11,21 .
(1) where B 0 is a constant. In order to ensure that the generated magnetic field is minimal, it is also expected that the liquid has weak electrical conductivity. The Rosseland approximation is used to simulate the flow of thermal radiation 11,31 . The heat flux is assumed to be proportionate to the temperature difference in this approximation, and heat moves from the hard surface to the liquid. In light of this, the radiative heat flux q r is written in the following form where σ * signifies Stefan-Boltzmann constant and k * symbolizes the rate. Assuming that T 4 can be represented as a linear function of temperature T 4 ≡ 4T 3 ∞ T − 3T 4 ∞ and that the temperature variations within the fluid flow are suitably low. With this, Eq. (6) can be written as Substituting Eq. (7) into Eq. (3) gives In order to simplify the governing equations of the present problem, the following specified similar quantities are introduced.
where η represents the similarity variable, f (η) is the dimensionless stream function, θ(η) corresponds to the non-dimensional temperature and the stream function ψ x, y satisfies the continuity Eq. (1) such that the components of the velocity uand v are defined as By using Eqs. (5) and (9), Eqs. (2), (4), and (8) can be transformed into the system of ordinary differential equations 1 www.nature.com/scientificreports/ The non-dimensional parameters that appeared in Eqs. (11)-(13) are the viscoelastic parameter K , magnetic parameter M , mixed convection parameter , Prandtl number Pr , radiation parameter R , and Eckert number Ec. These parameters are respectively defined as Upon formulation of the governing equations for mixed convective viscoelastic flow due to a nonlinearly stretching sheet with the aforementioned assumptions. Then, we develop the expressions for important physical parameters, which are of engineering interest. These parameters are the skin friction coefficient C f and the local Nusselt number Nu x . They provide examples of the stretched surface drag and rate of wall heat transfer. Then, the wall shear stress τ w is determined by The non-dimensional skin friction coefficient is defined by The skin friction coefficient in terms of transformation variables (9) and (10) can be obtained as The heat flux at the stretched surface may be calculated using and the heat transfer coefficient is defined as where Re = U w x ν is the local Reynolds number.
Numerical explanation procedure. In the current study, we reduced the dimensional governing equations into a non-dimensional form using similarity variables. The set of transformed differential Eqs. (11), (12) and corresponding boundary conditions (13) were numerically solved using Cebeci and Bradshaw's Keller box method 37 , an implicit finite difference technique. Selecting acceptable finite values of η → ∞ is the method's crucial stage. Nevertheless, we begin with starting values for pertinent parameters in order to establish f ′′ (0)and θ ′ (0) for the boundary value problem represented by Eqs. (10), (11) before attempting to determine η → ∞ . Until the findings are rectified to the necessary precision of 10 −5 level, the procedure is repeated. The drag force f ′′ (0) and Nusselt number θ ′ (0) were quantified numerically for each pair of sequential values. For a different set of physical factors, the values of η may vary. The related boundary value problem presented by Eqs. (10), (11) is then numerically addressed using the Keller box approach once the proper value of η has been established. This method was designed specifically for the parabolic type of differential equations. In this method, the numerical solution is obtained by considering the step size of 0.02 with a five-decimal places criterion of convergence for accuracy. In fact, one needs to consider the following four basic procedures in using the method: 1. Re-written the transformed Eqs. (10), (11) with their corresponding boundary conditions (12) into a firstorder system of equations. 2. Using the central difference method to convert the obtained first order equations into set finite difference equations. 3. Newton quasi-linearization method is then employed in solving the system of non-linear equations. 4. Block-matrix algorithm is finally used to solve the differential equations.

Results and discussion
To better comprehend the flow pattern, numerical figures for the temperature and velocity are generated for variations of non-dimensional factors, such as the nonlinear stretching parameter (n) , viscoelastic parameter (K), magnetic field parameter (M), mixed convection parameter ( ), radiation parameter (R), viscous dissipation parameter (Ec) and Prandtl number (Pr) . Additionally, the impact of these factors on non-dimensional Nusselt number and skin friction is explored. An investigation of the mixed convection flow of a viscoelastic fluid with combined effects of viscous dissipation and radiation across a nonlinearly extending sheet has been done in this paper. The whole flow fields are studied using the boundary layer idea in order to provide a set of linked ∂u ∂y ∂v ∂y y=0 .
(16) www.nature.com/scientificreports/ momentum and energy equations that may be used to solve the problem. The appropriate similarity variables have been provided, which converts the momentum and energy equations into a couple of nonlinear ordinary differential equations, hence simplifying the governing partial differential equations into ordinary differential equations. The Keller box approach was then used to achieve the numerical results. The analysis of the parameters n, K, M, , R, Ec, and Pr effects on the velocity and temperature distributions were inferred from the data. The impact of these factors on non-dimensional heat transfer coefficient and skin friction is also examined in tabular form. Comparing the numerical solutions of heat transfer rate −θ ′ (0) for various values of n and Pr by fixing K = M = = 0 and R = Ec = 0.1 with Hsiao 33 allowed for the determination of the numerical method's validity, as shown in Table 1. The comparison reveals good agreement, which supports the adoption of the Keller box approach.
Cooling is related to the heat transfer rate from the hot surface during numerous industrial manufacturing processes such as glass blowing, metal extrusion, glass fibre manufacture, and polymer extrusion. The primary goal in these types of issues is to limit the rate of heat transmission, which depends on the production method and the physical makeup of the issue at hand. Given this, Figs. 2, 3, 4, 5, 6, 7, 8 and 9, which illustrate the impacts of a viscoelastic, nonlinear stretching sheet, mixed convection, radiation parameter, Eckert and Prandtl numbers on the velocity f ′ (η) and temperature θ ′ (η) profiles are shown after the tables and their comments. Figures 2 and 3 are displayed to see the influence of viscoelastic parameter K on the velocity and temperature profiles. From this plot, it is evident that rising viscoelastic parameter K values cause the velocity across the boundary layer to rise. This implies that the fluid flow velocity increases as the viscoelastic parameter increases. This practically indicates that the characteristics of fluid flow in viscoelastic fluids may be influenced by modifying the change in the K . It is also obvious from Fig. 3 that to increase the magnitude of K is to decrease the temperature profile. When the dimensionless temperature profile is lower, K value is higher.
As a result, the fluid's heat may be readily removed from it by the viscoelasticity force, which has positive effects for more viscoelasticity. The reason for this is that while the thermal boundary layers are reduced as the viscoelastic effects become more pronounced, which results in a higher rate of heat transfer among the viscoelastic fluid and sheet surface, the material variables, which have an inverse relation with the fluid viscosity, are what causes the increase in a momentum boundary layer.
The behaviour of nonlinear stretching parameter n on velocity profile is seen in Fig. 4. It demonstrates this since an increase in n causes the stretching sheet to increase. In actuality, stretching velocity rises because n is increased, which causes the viscoelastic fluid to deform more. As n increases, the boundary layer thickness eventually thickens. Where n = 1 results correspond to linear stretching sheet and for n > 1 results correspond Table 1. Comparative analysis of −θ ′ (0) obtained by the numerical technique with that of Hsiao 33 for numerous values of n and Pr. Fixing K = M = = 0 and R = 0.2, Ec = 0.1.      www.nature.com/scientificreports/ to nonlinearly stretching surface. Studies comparing n = 1 and n > 1 show that when n > 1 (i.e., a nonlinearly stretching sheet) is present within the stretching boundary layer as opposed to the linear stretching sheet situation, velocity magnitude considerably rises. The behaviour of nonlinear stretching parameter n on temperature profile is depicted in Fig. 5. Practically, it is known that the heat transfer from a thinner region will be greater than that of a thicker region. The stretching rate of the surface rises as n grows because more energetic particles are delivered to the flow; as a response, the thickness of the sheet decreases, which improves the rate of heat transmission. Significantly, the thermal boundary layer gets thinner for larger values of n. Figure 6 illustrates the effects of mixed convection parameters on the velocity profile. It is revealed that strengthening the value of enhances the buoyancy effect in the flow regime and hence, the velocity profile grows. The buoyancy-assist flow ( > 0) has a higher velocity profile than the buoyancy-resisting flow ( < 0) , and vice versa. This effect is caused physically by buoyancy force, which behaves as a pressure gradient and propels or depreciates the fluid in the boundary layer, in turn raising the velocity profile. The effects on the temperature profile show that it has a minimal impact on the temperature profile, hence no graph is presented. Figure 7 shows the characteristics of radiation parameter R with the temperature profile. It is observed from this graph that the thermal field increased with the increase in R . More heat is transferred to the working liquid via the radiation phenomena because the radiation parameter is inversely related to the mean absorption coefficient. A higher radiation parameter value causes k * to decline, which raises the temperature. As a result, the rate of thermal convection into the fluid increases, as does the deviation of radiative heat flow. The increased thermal heat transfer is advantageous for the development of thermal boundary layers. As a result, the thermal layer thickens.
Whether the sheet is being heated or cold affects how important viscous dissipation is. Due to the viscoelastic nature of the fluid, we used for the analysis, frictional heating caused by viscous dissipation will be used to store the energy in the fluid. Figure 8 displays the variation of viscous dissipation which is represented in terms of Eckert number Ec on the temperature profile. When a fluid is heated, viscous dissipation (Ec > 0) causes the temperature to rise, however when a fluid is cooled, we see the opposite behaviour regarding the temperature profile when (Ec < 0) . On the other hand, due to the effect of viscous dissipation, the existence of viscous dissipation in the energy equation works as an internal heat source, causing the dimensionless temperature to increase at Ec > 0 in comparison to Ec < 0 . Due to frictional heating, Ec produces more heat in the liquid when it is physically bigger. Consequently, a hike in temperature is seen.
The trend of the Prandtl number Pr on the temperature profile is seen in Fig. 9. Fluids experience both conduction and convection. They can be viewed as competing with one another in terms of transmitting heat because the temperature differential is decreased by both processes, which transfer heat. Fluids come in a variety of forms, including mercury, oil, water, air, and many others. Various fluids exhibit different rates of conduction and convection. Conduction can sometimes take control. Convection occasionally takes control. One metric that can help indicate which process will prevail is the Prandtl number. If a fluid is more viscous, the Prandtl number is higher and the heat transfer will be less convective since Pr is the ratio of momentum to thermal thermal diffusivity. The thermal boundary layer thickness and temperature distribution are reduced as a result of poorer thermal diffusivity, which is associated with larger values of Pr.
For various values of the magnetic parameter M and viscoelastic parameter K , the values of the skin friction coefficient are shown in Table 2. The numbers in the table show that increasing the values of the skin friction coefficient is a result of strengthening the values of M . This implies that when the magnetic field's strength increases, the fluid flow will face significant resistance or drag. This happens because a higher magnetic field causes a Lorenz drag, which raises the value of the drag force. In practice, this means that raising the intensity of the magnetic field might be used to limit the fluid flow rate, which could be a beneficial method for managing the final qualities of the extrusion properties. Similar to this, it is observed that when the values of the viscoelastic parameters K grow, www.nature.com/scientificreports/ the magnitude values of −f ′′ (0) decrease. Since the power required to stretch a sheet is reduced by strengthening the value K , this result is significant in many industrial applications. Table 3 displays how the non-dimensional heat transfer coefficient −θ(0) is affected by magnetic parameter M , radiation parameter R , and viscous dissipation parameter Ec . From the data in the table, it is clear that decreasing the value of any one of the three factors would decrease the value of the heat transfer coefficient. As a result, the heat transfer coefficient will decrease as the magnetic field's intensity increases, the fluid flow regime is exposed to radiative heat flux, and thermal energy from frictional heating caused by fluid movement is taken into account. This happens as a result of a sluggish heat transfer from the wall to the fluid caused by the magnetic field's reduction in fluid flow rate. The temperature of the fluid rises as a result of increased thermal radiation and viscous dissipation, which also slows heat transmission from the wall to the fluid. By strengthening the magnetic field and raising the fluid's temperature, it is possible to practically regulate the heat transfer across the stretched sheet's fluid interface. To help create a better cooling environment, radiation should be reduced to a minimum.

Conclusion
In the presence of an applied magnetic field, thermal radiation, and viscous dissipation, the mathematical model including mixed convection boundary layer flow of viscoelastic fluid and heat transfer phenomena across a nonlinearly stretching sheet has been investigated. When building heat transfer systems for diverse industrial applications, the rate of heat transmission is a key consideration. Depending on the applications, it may occasionally be necessary to raise or decrease the heat transfer rate to achieve better outcomes in the heating or cooling operations. Additionally, the findings are important since the material qualities of the wall rely on the pace of cooling or heating during the production of metal or polymer sheets. Therefore, an increase in temperature distribution and the associated thermal layer is caused by radiative and viscous dissipation. In reality, radiation causes more heat to be produced in liquids, raising their temperature. Radiation has an impact on heat transport and temperature distribution at high temperatures. Within the nonlinearly expanding boundary layer, opposing